Propagation Velocity Calculator

Enter lag observations

Use distance-time pairs from observed propagation events. The calculator estimates individual speeds, summary statistics, fitted velocity, and uncertainty measures.

Distance unit

Time unit

Confidence level

Measurement error per distance observation

Expected velocity for z comparison

Interpretation note

Observation	Distance	Time Delay	Weight
1
2
3
4
5
6

Formula used

1) Pairwise propagation velocity
v_i = d_i / t_i

2) Arithmetic mean velocity
Mean = Σv_i / n

3) Weighted mean velocity
Weighted Mean = Σ(w_i × v_i) / Σw_i

4) Sample standard deviation
s = √[Σ(v_i - mean)² / (n - 1)]

5) Standard error
SE = s / √n

6) Confidence interval
CI = mean ± z × SE

7) Fitted velocity from regression through origin
Fitted Velocity = Σ(t_i × d_i) / Σ(t_i²)

This design suits statistical lag analysis, event diffusion studies, wavefront tracking, or spatial-temporal propagation work where distance grows with delay.

How to use this calculator

Enter the distance unit and time unit.
Select a confidence level for interval reporting.
Add at least two valid distance-time observations.
Optionally set weights for more trusted measurements.
Enter expected velocity for comparison testing.
Click the calculate button to view summaries.
Review the result banner above the form.
Study the detailed table and fitted trend graph.
Download CSV or PDF for reporting needs.

Example data table

Observation	Distance	Time Delay	Weight	Velocity
1	120	2.4	1.0	50.00
2	250	4.9	1.0	51.02
3	390	7.8	1.0	50.00
4	520	10.1	1.0	51.49

These example values show how similar lag observations can produce a stable central velocity estimate.

FAQs

What is a propagation velocity calculator?

It estimates how quickly an event, signal, effect, or pattern moves across distance over time. This version also summarizes variability, confidence limits, and fitted trend strength.

Why are multiple observations better than one?

Multiple observations reveal consistency and spread. A single pair only gives one speed, while several pairs support mean estimates, uncertainty ranges, and regression-based validation.

What does the fitted velocity mean?

Fitted velocity is the slope from distance versus time using regression through the origin. It represents the overall propagation rate implied by the whole dataset.

When should I use weights?

Use weights when some measurements are more reliable, precise, or representative than others. Larger weights give those observations more influence in the weighted mean velocity.

What does the confidence interval show?

The interval shows a plausible range for the mean propagation velocity, based on sample spread and chosen confidence level. Wider intervals indicate greater uncertainty.

What is the coefficient of variation?

It expresses standard deviation relative to the mean as a percentage. Lower values suggest more stable propagation estimates across the observed lag pairs.

Why compare with an expected velocity?

Comparing with an expected value helps check whether observed propagation appears faster or slower than a benchmark, model assumption, or historical reference rate.

Can I use different units?

Yes. Enter any consistent distance and time units, such as kilometers and hours or meters and seconds. The output velocity follows the same unit ratio.